Graph \(\sigma _3(G)\)  is a RAC graph by
Theorem REF (ii). It has at most \(3m\)  edges, so that
there are at most \(3km\)  crossings in a \(k\) -gap-planar drawing of
\(\sigma _3(G)\) . Since the crossing number is preserved by
subdivisions, \(\sigma _3(G)\)  is not \(k\) -gap-planar if \(cr(G) > 3km\) .
Then it is not \(2k\) -planar by [1]}.
